mediation: R package for causal mediation analysis

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August 2014, Volume 59, Issue 5. http://www.jstatsoft.org/

mediation: R Package for Causal Mediation Analysis

Dustin Tingley Harvard University Teppei Yamamoto Massachusetts Institute of Technology Kentaro Hirose Princeton University Luke Keele

Pennsylvania State University

Kosuke Imai

Princeton University

Abstract

In this paper, we describe the R package mediation for conducting causal mediation analysis in applied empirical research. In many scientific disciplines, the goal of researchers is not only estimating causal effects of a treatment but also understanding the process in which the treatment causally affects the outcome. Causal mediation analysis is fre-quently used to assess potential causal mechanisms. The mediation package implements a comprehensive suite of statistical tools for conducting such an analysis. The package is organized into two distinct approaches. Using the model-based approach, researchers can estimate causal mediation effects and conduct sensitivity analysis under the standard research design. Furthermore, the design-based approach provides several analysis tools that are applicable under different experimental designs. This approach requires weaker assumptions than the model-based approach. We also implement a statistical method for dealing with multiple (causally dependent) mediators, which are often encountered in practice. Finally, the package also offers a methodology for assessing causal mediation in the presence of treatment noncompliance, a common problem in randomized trials.

Keywords: causal mechanisms, mediation analysis, mediation, R.

1. Introduction

Scholars across a wide range of disciplines are increasingly interested in identifying causal mechanisms, going beyond the estimation of causal effects. Once they ascertain that cer-tain variables causally affect the outcome, the next natural step is to understand how these variables exert their influence. The standard procedure for analyzing causal mechanisms in applied research is called mediation analysis, where a set of linear regression models are

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fit-ted and then the estimates of “mediation effects” are compufit-ted from the fitfit-ted models (e.g.,

Haavelmo 1943; Baron and Kenny 1986; Shadish, Cook, and Campbell 2001; MacKinnon

2008). In recent years, however, causal mechanisms have been studied within the modern

framework of causal inference with an emphasis on the assumptions required for identifi-cation. This approach has highlighted limitations of earlier methods and pointed the way towards a more flexible estimation strategy. In addition, new research designs have been proposed for identifying causal mechanisms.

In this paper, we introduce a full featured R package, mediation (Tingley, Yamamoto, Hirose,

Keele, and Imai 2013), for studying causal mechanisms. The mediation package allows users to (1) investigate the role of causal mechanisms using different types of data and statistical models, (2) explore how results change as identification assumptions are relaxed, and (3) calculate quantities of interest under alternative research designs. We focus on the demon-stration of the functionalities available through the mediation package. The statistical theory that underlies the procedures implemented in the mediation package is presented elsewhere

along with various empirical examples (Imai, Keele, and Yamamoto 2010c;Imai, Keele,

Tin-gley, and Yamamoto 2011; Imai, Keele, and Tingley 2010a; Imai, Tingley, and Yamamoto 2013;Yamamoto 2013).

The mediation package is freely available for download via the Comprehensive R Archive

Net-work (CRAN) at http://CRAN.R-project.org/package=mediation and runs on a variety

of computing platforms (R Core Team 2014). In addition, a Stata (StataCorp. 2013) version

of the package is available but has a more limited functionality (Hicks and Tingley 2011). The

first version of the mediation package appeared at CRAN in 2009, andImai, Keele, Tingley,

and Yamamoto (2010b) discuss an earlier version of the package. Since then, however, we have dramatically improved the package with a significant number of new functionalities and improvements. The current paper thus provides an up-to-date description of the analyses that can be conducted via the mediation package. To install the mediation package, use the following standard syntax for installing an R package,

R> install.packages("mediation")

where users may be prompted to select a CRAN mirror from which the package will be downloaded. This step needs to be done only once (unless one wishes to update the mediation package to the new version).

In the next section, we present an overview of the mediation package. We then describe the

functionalities of the package for the model-based causal mediation analysis (Section3),

mul-tilevel mediation analysis (Section4), the design-based causal mediation analysis (Section5),

the analysis of causally dependent multiple mediators (Section6), and causal mediation

anal-ysis with treatment noncompliance (Section7). Finally, Section8concludes.

2. Overview of the mediation package

The mediation package consists of several main functions as well as various methods for summarizing output from these functions (e.g., plot and summary). The package requires

little programming knowledge on the user’s side. Figure 1 illustrates the core structure of

the mediation package, which distinguishes between model-based and design-based inference. Model-based inference has been standard practice in the mediation analysis to date. In the

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Figure 1: Core structure of the mediation package as of version 4.0.

experimental setting, the treatment variable is randomized and the mediating and outcome

variables are observed without any intervention by researchers. Imai et al.(2010a) show that

a range of parametric and semi-parametric models may then be used to estimate the average causal mediation effect, defined below, and other quantities of interest. This modeling

ap-proach relies on the sequential ignorability assumption for point identification, which asImai

et al.(2010a) show, provides a general purpose algorithm for estimating quantities of interest. In contrast, design-based inference primarily employs the features of the experimental design and does not require the sequential ignorability assumption. The formal identification

prop-erties of these designs are studied byImai et al. (2013) and the examples from experimental

and observational studies are contained inImai et al.(2011,2013). We refer readers to these

papers for the details about the statistical methods implemented via the mediation package. Before describing the functions available in mediation, we briefly define the quantities of inter-est that our software is designed to inter-estimate. Here, we use the potential outcomes framework

to define these quantities. Let Mi(t) denote the potential value of a mediator of interest for

unit i under the treatment status Ti = t. Let Yi(t, m) denote the potential outcome that

would result if the treatment and mediating variables equal t and m, respectively. Consider a standard experimental design where only the treatment variable is randomized. We observe only one of the potential outcomes, and the observed outcome, Yi, equals Yi(Ti, Mi(Ti)) where Mi(Ti) represents the observed value of the mediator Mi. With this notation, the total unit treatment effect can be written as,

τi ≡ Yi(1, Mi(1)) − Yi(0, Mi(0)). (1)

We can decompose this total effect into the two components. First, the causal mediation

effects are represented by (Robins and Greenland 1992;Pearl 2001),

δi(t) ≡ Yi(t, Mi(1)) − Yi(t, Mi(0)), (2)

for each treatment status t = 0, 1. All other causal mechanisms can be represented by the direct effects of the treatment as,

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for each unit i and each treatment status t = 0, 1. Together, we see that they sum up to the total effect,

τi = δi(t) + ζi(1 − t) (4)

for t = 0, 1. The case of multiple candidate mediating variables requires additional notation

and is discussed in Section 6. The average causal mediation effects (ACME) ¯δ(t) and the

average direct effects (ADE) ¯ζ(t), represent the population averages of these causal mediation

and direct effects.

Identification of the ACME requires an additional assumption beyond the strong ignorability of the treatment, which is sufficient for identifying the average total effect of the treatment.

Let Xi be a vector of the observed pre-treatment confounders for unit i. The key identifying

assumption is called sequential ignorability and can be written as,

Assumption 1 (Sequential Ignorability; Imai et al. 2010c)

{Yi(t0, m), Mi(t)} ⊥⊥ Ti | Xi = x, (5)

Yi(t0, m) ⊥⊥ Mi(t) | Ti = t, Xi= x, (6)

where 0 < P(Ti = t | Xi = x) and 0 < p(Mi = m | Ti = t, Xi = x) for t = 0, 1, and all x and

m in the support of Xi and Mi, respectively.

Equation5is the standard strong ignorability of the treatment assignment and is satisfied, for

example, if the treatment is randomized (possibly conditional on Xi). However, Equation 6

requires that the mediator is also ignorable given the observed treatment and pre-treatment confounders. This additional assumption is quite strong because it excludes the existence of (measured or unmeasured) post-treatment confounders as well as that of unmeasured pre-treatment confounders. This assumption, therefore, rules out the possibility of multiple

me-diators that are causally related to each other (see Section6 for the method that is designed

to deal with such a scenario).

3. Model-based causal mediation analysis

In this section, we discuss the functionalities of the mediation package for model-based causal mediation analysis under the assumption of sequential ignorability. Many of these

function-alities are described in detail in Imai et al. (2010b), but the current version of the package

accommodates a larger class of statistical models.

The model-based causal mediation analysis proceeds in two steps. First, the researcher speci-fies two statistical models, the mediator model for the conditional distribution of the mediator

Mi given the treatment Ti and a set of the observed pre-treatment covariates Xi and the

out-come model for the conditional distribution of the outout-come Yi given Ti, Mi, and Xi. These

models are fitted separately and then their fitted objects comprise the main inputs to the mediate function, which computes the estimated ACME and other quantities of interest un-der these models and the sequential ignorability assumption. Since the sequential ignorability assumption is untestable, we recommend that the researchers conduct a sensitivity analy-sis via the medsens function, which can be applied to certain statistical models. We now illustrate these functionalities with an empirical example.

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Outcome model types

Mediator model types Linear GLM Ordered Censored Quantile GAM Survival

Linear (lm/lmer) X X X∗ X X X∗ X

GLM (glm/bayesglm/ X X X∗ X X X∗ X

glmer)

Ordered (polr/bayespolr) X X X∗ X X X∗ X

Censored (tobit via vglm) – – – – – – –

Quantile (rq) X∗ X∗ X∗ X∗ X∗ X∗ X

GAM (gam) X∗ X∗ X∗ X∗ X∗ X∗ X∗

Survival (survreg) X X X∗ X X X∗ X

Table 1: Types of statistical models that can be used with the mediate function.

Aster-isks,∗, indicate the model combinations that can only be estimated using the nonparametric

bootstrap (i.e., with the argument boot = TRUE for the mediate function).

3.1. Estimation of the average causal mediation effects

The mediate function takes various standard model objects (such as obtained with lm and glm), which correspond to mediator and outcome models, and returns the estimates of the average causal mediation effects along with other causal quantities of interest. The output of the mediate function can be passed to the plot and summary functions in order to obtain graphical and numerical summaries, respectively. The mediate function automatically detects the type of models used for the mediator and outcome models and calculates the estimates of

the ACME and other quantities of interest via the general algorithms described inImai et al.

(2010a). Our estimation strategy overcomes the limitation of the standard methods based on the product or difference of coefficients, which are only appropriate for the analysis of causal mediation effects when both the mediator and outcome models are linear regressions where

Ti and Mi enter the models additively (e.g., without interaction). In contrast, the algorithms

used in the mediation package nest this as a special case and accommodate a greater range

of statistical models as shown in Table1.

We now illustrate the use of the mediate function with an empirical example based on the

framing data ofBrader, Valentino, and Suhat(2008). This data set is a part of the mediation

package and can be loaded via the following syntax, R> library("mediation")

R> set.seed(2014)

R> data("framing", package = "mediation")

A brief description of each variable in the data can be obtained through a help file, R> ?framing

Brader et al.(2008) conducted a randomized experiment where subjects are exposed to differ-ent media stories about immigration and the authors investigated how their framing influences attitudes and political behavior regarding immigration policy. They posit anxiety as the me-diating variable for the causal effect of framing on public opinion. We first fit the mediator model where the measure of anxiety (emo) is modeled as a function of the framing treatment

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(treat) and pre-treatment covariates (age, educ, gender, and income). Next, we model the outcome variable, which is a binary variable indicating whether or not the participant agreed to send a letter about immigration policy to his or her member of Congress (cong_mesg). The explanatory variables of the outcome model include the mediator, treatment status, and the

same set of pre-treatment variables as those used in the mediator model.1 In this example,

the treatment is expected to increase the level of respondents’ emotional response, which in turn is hypothesized to make subjects more likely to send a letter to his or her member of Congress. We use the linear regression fit with least squares and the probit regression for the mediator and outcome models, respectively.

R> med.fit <- lm(emo ~ treat + age + educ + gender + income, data = framing) R> out.fit <- glm(cong_mesg ~ emo + treat + age + educ + gender + income,

+ data = framing, family = binomial("probit"))

We now use the mediate function to estimate the ACME and ADE. As the inputs to this function, we must specify the model fits (in this case med.fit and out.fit) as well as the names of the treatment and mediating variables, which are represented as the arguments treat and mediator, respectively. Here and throughout the rest of this paper, we use a small number of simulations (sims = 100) for the purpose of illustration to calculate the uncertainty estimates, but one may wish to use the default (1000) or even larger number if the estimates vary too much from one simulation to another. The default simulation type is

the quasi-Bayesian Monte Carlo method based on normal approximation (Imai et al. 2010a).

We use White’s heteroskedasticity-consistent estimator for the covariance matrix from the

sandwich package (vcovHC; Zeileis 2006) by setting the robustSE argument to TRUE. This

argument can be omitted if standard uncertainty estimates are desired. Finally, like most functions in R, the results of the mediate function can be summarized numerically by the summary function, which calculates point estimates, confidence intervals, and the p-values, for

the average direct, indirect, and total effects.2 The syntax is now given as,

R> med.out <- mediate(med.fit, out.fit, treat = "treat", mediator = "emo",

+ robustSE = TRUE, sims = 100)

R> summary(med.out)

Causal Mediation Analysis

Quasi-Bayesian Confidence Intervals

Estimate 95% CI Lower 95% CI Upper p-value

ACME (control) 0.0791 0.0351 0.1501 0.00

ACME (treated) 0.0804 0.0367 0.1557 0.00

ADE (control) 0.0206 -0.0976 0.1158 0.70

ADE (treated) 0.0218 -0.1053 0.1226 0.70

1

Using different sets of pre-treatment covariates for the mediator and outcome models may be justified depending on the causal relationships assumed for those covariates. SeePearl(2014) andImai, Keele, Tingley,

and Yamamoto(2014).

2_{Note that the results will be slightly different in each run of mediate because of Monte Carlo errors,} especially when sims is set to a small number. If exact reproduction of results is desired, users can set a specific randomness seed (set.seed) before calling the mediate function.

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Total Effect 0.1009 -0.0497 0.2339 0.14

Prop. Mediated (control) 0.6946 -6.3109 3.6793 0.14

Prop. Mediated (treated) 0.7118 -5.7936 3.4965 0.14

ACME (average) 0.0798 0.0359 0.1537 0.00

ADE (average) 0.0212 -0.1014 0.1192 0.70

Prop. Mediated (average) 0.7032 -6.0523 3.5879 0.14

Sample Size Used: 265 Simulations: 100

One new feature in the tabular output from the mediate functions is the addition of p-values for the various estimates. In this example, the estimated ACMEs are statistically significantly different from zero but the estimated average direct and total effects are not. The results suggest that the treatment in the framing experiment may have increased emotional response, which in turn made subjects more likely to send a message to his or her member of Congress. Here, since the outcome is binary all estimated effects are expressed as the increase in probability that the subject sent a message to his or her Congress person.

In addition, we can use the nonparametric bootstrap rather than the quasi-Bayesian Monte Carlo simulation for variance estimation via the boot = TRUE argument,

R> med.out <- mediate(med.fit, out.fit, boot = TRUE, treat = "treat",

+ mediator = "emo", sims = 100)

R> summary(med.out)

Nonparametric Bootstrap Confidence Intervals with the Percentile Method Estimate 95% CI Lower 95% CI Upper p-value

ACME (control) 0.0832 0.0426 0.1332 0.00

ACME (treated) 0.0844 0.0425 0.1333 0.00

ADE (control) 0.0114 -0.1158 0.1277 0.84

ADE (treated) 0.0125 -0.1274 0.1360 0.84

Total Effect 0.0958 -0.0477 0.2171 0.24

ACME (average) 0.0838 0.0434 0.1319 0.00

ADE (average) 0.0120 -0.1210 0.1318 0.84

The output now indicates that the bootstrap is used for inferences. As expected, the results are largely the same. In general, as long as computing power is not an issue, analysts should

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−0.1 0.0 0.1 0.2 ● ● ● ● ● Total Effect ADE ACME

Figure 2: Graphical display of results from the mediate function.

estimate confidence intervals via the bootstrap with more than 1000 resamples, which is the default number of simulations.

Two types of methods for calculating bootstrap-based confidence intervals are available via the boot.ci.type argument. The basic percentile intervals are calculated by default or setting the argument to "perc". The bias-corrected and accelerated (BCa) intervals are computed if the argument is set to "bca" (seeDiCiccio and Efron 1996, for the definition of the method). The latter has better asymptotic properties and is often recommended for the estimation of

mediation effects (Preacher and Hayes 2008).

As an alternative to the numerical summary, using the output from the mediate function as the input to the plot command provides a graphical summary of the three parameters

(indirect, direct, and total effects) along with their confidence intervals. Figure 2 shows the

result of plotting the med.out object.3

Treatment and mediator interaction

It is possible that the ACME takes different values depending on the baseline treatment status. In such a situation, the researcher can add an interaction term between the treatment and mediator to the outcome model. Then, the mediate function automatically detects the change

in the specification and explicitly estimates the ACME conditional on treatment status.4 In

the output given below, the estimated ACME now varies with treatment status.

R> med.fit <- lm(emo ~ treat + age + educ + gender + income, data = framing) 3

Users may make further modifications to the plot via standard plot options, including changes to the labels.

4_{When the outcome model is nonlinear, the ACME and direct effect estimates will differ between the} treatment and control conditions even when the model does not include an interaction term. The summary output in such cases includes average values of these two estimates to ease interpretation of the results.

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R> out.fit <- glm(cong_mesg ~ emo * treat + age + educ + gender + income,

R> summary(med.out)

ACME (control) 0.07942 0.02497 0.14275 0.00

ACME (treated) 0.10362 0.03558 0.17073 0.00

ADE (control) 0.00319 -0.10976 0.13230 0.98

ADE (treated) 0.02739 -0.11584 0.16657 0.68

Total Effect 0.10682 -0.05053 0.24410 0.20

ACME (average) 0.09152 0.03203 0.14967 0.00

ADE (average) 0.01529 -0.11744 0.14746 0.90

The statistical significance of the treatment-mediator interaction can be tested via the test.TMint function in the following manner.

R> test.TMint(med.out, conf.level = .95) Test of ACME(1) - ACME(0) = 0

data: estimates from med.out

ACME(1) - ACME(0) = 0.0242, p-value = 0.3

alternative hypothesis: true ACME(1) - ACME(0) is not equal to 0 95 percent confidence interval:

-0.01795541 0.06809042

The mediate function’s output contains a range of additional quantities that users might find helpful. Each is stored as part of the model’s output. This includes vectors of the simulation output for all quantities of interests (see ?mediate for details), which can be used for a variety of tasks, such as more intensive plotting.

Missing data

Our simulation-based approach to the estimation of mediation effects enables users to deal with missing data via standard multiple imputation procedures in a straightforward fashion.

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The mediation package includes a pair of utility functions – mediations and amelidiate – to facilitate such analysis. First, users simulate multiple data sets using their preferred imputation software. Next, run mediate on each data set by simply passing the data sets through mediations. Next, pass the output of mediations to the amelidiate function, which combines the components of the output from mediations into a format that can be

analyzed with the standard summary and plot commands.5 Alternatively, users can manually

run mediate on their imputed data sets and simply stack the vectors of quantities they are interested in, and use basic functions like quantile to calculate confidence intervals.

3.2. Moderated mediation

One new important feature of the mediate function is the ability to study moderated me-diation. Often analysts hypothesize that the magnitude of the ACME depends on (or is moderated by) a pre-treatment covariate. Such a pre-treatment covariate is called a moder-ator. In the framing example, the ACME may be much stronger among older respondents than younger ones. In other words, the ACME may be moderated by age.

There are two alternative routes to the analysis of moderated mediation with the mediation package. The first method involves alteration of both the statistical models as well as the syntax for the mediate function. First, the mediator and outcome models should contain the moderator and its interaction terms with respect to the treatment and mediating variables that are theoretically justified. For example, we may modify the previous models as follows, R> med.fit <- lm(emo ~ treat * age + educ + gender + income, data = framing) R> out.fit <- glm(cong_mesg ~ emo + treat * age + emo * age + educ + gender +

+ income, data = framing, family = binomial("probit"))

Once the two models are fitted, the researcher must specify the levels of the moderator at

which effects will be calculated by the mediate function.6 _{In the current example, this can}

be done by setting the age covariate to a specific value. To allow the mediation effects to be moderated by age, we set the value of age to be 20 in one model and 60 in another model. More complicated moderated mediation involving multiple moderators could be specified by expanding the list of the covariates.

R> med.age20 <- mediate(med.fit, out.fit, treat = "treat",

+ mediator = "emo", covariates = list(age = 20), sims = 100)

R> med.age60 <- mediate(med.fit, out.fit, treat = "treat",

+ mediator = "emo", covariates = list(age = 60), sims = 100)

R> summary(med.age20) Causal Mediation Analysis

Quasi-Bayesian Confidence Intervals 5

Note that amelidiate does not support some models and features yet; see ?amelidiate for details. 6_{If the models include moderator-treatment interactions and yet this option is not specified, then the} resulting ACME and direct effect estimates will simply be averages over the empirical distribution of the covariates.

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(Inference Conditional on the Covariate Values Specified in `covariates') Estimate 95% CI Lower 95% CI Upper p-value

ACME (control) 0.0702 0.0101 0.1813 0.04

ACME (treated) 0.0852 0.0144 0.2020 0.04

ADE (control) 0.2275 0.0224 0.4638 0.04

ADE (treated) 0.2425 0.0248 0.4714 0.04

Total Effect 0.3127 0.1122 0.5568 0.00

Prop. Mediated (control) 0.2126 0.0235 0.8238 0.04

Prop. Mediated (treated) 0.2641 0.0334 0.8608 0.04

ACME (average) 0.0777 0.0123 0.1914 0.04

ADE (average) 0.2350 0.0236 0.4676 0.04

Prop. Mediated (average) 0.2383 0.0285 0.8423 0.04

Sample Size Used: 265 Simulations: 100 R> summary(med.age60) Causal Mediation Analysis

(Inference Conditional on the Covariate Values Specified in `covariates')

ACME (control) 0.07703 0.01058 0.13799 0.04

ACME (treated) 0.06900 0.00919 0.13829 0.04

ADE (control) -0.08905 -0.22558 0.05295 0.28

ADE (treated) -0.09708 -0.24478 0.05592 0.28

Total Effect -0.02005 -0.17471 0.14057 0.78

Prop. Mediated (control) -0.52540 -8.25181 17.47875 0.78

Prop. Mediated (treated) -0.43131 -7.16792 16.01512 0.78

ACME (average) 0.07302 0.00989 0.13905 0.04

ADE (average) -0.09306 -0.23236 0.05453 0.28

Prop. Mediated (average) -0.47836 -7.70987 16.74694 0.78

Thus, the researcher receives two different sets of output. In the first output, the average mediation effect is estimated for those who are 20 years old. In contrast, the second output applies to those who are 60 years old.

The second approach to moderated mediation consists of directly testing the statistical sig-nificance of the difference in the ACME and ADE between two chosen levels of pre-treatment

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covariates. This analysis is conducted via the test.modmed function. For example, the fol-lowing syntax can be used to test whether the ACME and ADE significantly differ between the subjects who are 20 years old and those who are 60 years old.

R> med.init <- mediate(med.fit, out.fit, treat = "treat", mediator = "emo",

+ sims = 2)

R> test.modmed(med.init, covariates.1 = list(age = 20),

+ covariates.2 = list(age = 60), sims = 100)

Test of ACME(covariates.1) - ACME(covariates.2) = 0

data: estimates from med.init

ACME(covariates.1) - ACME(covariates.2) = 0.008, p-value = 0.92

alternative hypothesis: true ACME(covariates.1) - ACME(covariates.2) is not equal to 0

95 percent confidence interval:

-0.1075738 0.1249199

Test of ADE(covariates.1) - ADE(covariates.2) = 0

data: estimates from med.init

ADE(covariates.1) - ADE(covariates.2) = 0.3027, p-value = 0.02

alternative hypothesis: true ADE(covariates.1) - ADE(covariates.2) is not equal to 0

95 percent confidence interval: 0.04676954 0.59796646

Unlike the first approach, the initial mediate fit does not need the covariates argument set to specific values; the choice of covariate levels is made directly in the call to the test.modmed function. Note that the initial mediate call does not require a large number of simulation draws, for the actual calculation of uncertainty for the test happens within the test.modmed function.

3.3. Non-binary treatment variables

Experimental manipulations are often complex and go beyond simple treatment and control conditions. In the framing experiment, for example, the researchers actually used a 2 × 2 factorial design. That is, each subject was exposed to two different binary treatments, yielding four different experimental manipulations. In the analysis presented above, we have focused on a comparison of one of these conditions relative to the other three conditions. The mediate function, however, has the capability to handle more complex experimental contrasts, which can be represented by a non-binary treatment variable.

Here, instead of using the binary treat variable, we use a variable named cond, which records which of the four conditions the subject was randomly exposed to. Using the control.value and treat.value options, the user can calculate the specific contrast of interest. For example, the comparison between the second and third conditions can be done with the following code.

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Outcome model types

Mediator model types Linear Binary probit

Linear X X

Binary probit X –

Table 2: The types of models that can be handled by medsens for sensitivity analysis.

R> med.fit <- lm(emo ~ cond + age + educ + gender + income, data = framing) R> out.fit <- glm(cong_mesg ~ emo + cond + age + educ + gender + income,

R> med23.out <- mediate(med.fit, out.fit, treat = "cond", mediator = "emo",

+ control.value = 2, treat.value = 3, sims = 100)

R> summary(med23.out)

Similarly, the researcher can compare the first and fourth experimental conditions via the following syntax,

R> med14.out <- mediate(med.fit, out.fit, treat = "cond", mediator = "emo",

R> summary(med14.out)

Nothing changes in the format of the output, but the contrasts differ depending on the categories chosen for comparison by the researcher. In the case of a continuous treatment

variable, the researcher would specify two values of the treatment to make the contrast (Imai

et al. 2010a). For example, the causal mediation effects can be defined for any two levels of the treatment,

δi(t; t1, t0) ≡ Yi(t, Mi(t1)) − Yi(t, Mi(t0)), (7)

where t1 6= t0. The corresponding average causal mediation effect is defined as ¯δ(t; t1, t0) ≡

E(δi(t; t1, t0)). Thus, the researcher can set control.value to t0 and treat.value to t1. The

researcher may also vary the value of t1, while fixing the base line value of t0, to examine how

the ACME changes as the function of t1.

3.4. Sensitivity analysis for sequential ignorability

Sequential ignorability is a strong assumption, and therefore a sensitivity analysis is recom-mended. The mediation package allows the researcher to conduct a sensitivity analysis for the possible existence of unobserved pre-treatment covariates. Specifically, the output of the mediate function can be passed to the medsens function, which then computes the values of causal quantities as a function of sensitivity parameters. Both summary and plot functions are available for sensitivity analysis, and they display the results in a tabular and graphical form, respectively. Since derivation of sensitivity formulas must be done on a case-by-case

basis, the range of options for conducting sensitivity analyses is somewhat limited. Table 2

gives the model combinations currently supported by the medsens function.

In our running example, after computing the ACME, we conduct a sensitivity analysis by passing the object from mediate to the medsens function. We first choose as the sensitivity parameter the correlation ρ between the residuals of the mediator and outcome regressions

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(Imai et al. 2010a,c). If there exist unobserved pre-treatment confounders which affect both the mediator and the outcome, we expect that the sequential ignorability assumption is vio-lated and ρ is no longer zero. The sensitivity analysis is conducted by varying the value of ρ and examining how the estimated ACME changes. The following syntax can be used to conduct this analysis,

R> med.fit <- lm(emo ~ treat + age + educ + gender + income, data = framing) R> out.fit <- glm(cong_mesg ~ emo + treat + age + educ + gender + income,

R> sens.out <- medsens(med.out, rho.by = 0.1, effect.type = "indirect",

+ sims = 100)

R> summary(sens.out)

Mediation Sensitivity Analysis: Average Mediation Effect Sensitivity Region: ACME for Control Group

Rho ACME(control) 95% CI Lower 95% CI Upper R^2_M*R^2_Y* R^2_M~R^2_Y~

[1,] 0.3 0.0058 -0.0055 0.0206 0.09 0.0493

[2,] 0.4 -0.0095 -0.0285 0.0024 0.16 0.0877

Rho at which ACME for Control Group = 0: 0.3

R^2_M*R^2_Y* at which ACME for Control Group = 0: 0.09 R^2_M~R^2_Y~ at which ACME for Control Group = 0: 0.0493

Sensitivity Region: ACME for Treatment Group

Rho ACME(treated) 95% CI Lower 95% CI Upper R^2_M*R^2_Y* R^2_M~R^2_Y~

[1,] 0.3 0.0066 -0.0069 0.0222 0.09 0.0493

[2,] 0.4 -0.0118 -0.0351 0.0026 0.16 0.0877

Rho at which ACME for Treatment Group = 0: 0.3

R^2_M*R^2_Y* at which ACME for Treatment Group = 0: 0.09 R^2_M~R^2_Y~ at which ACME for Treatment Group = 0: 0.0493

where rho.by = 0.1 specifies that ρ will vary from −0.9 to 0.9 by 0.1 increments, and effect.type = "indirect" means that sensitivity analysis is conducted for the ACME. Al-ternatively, specifying effect.type = "direct" performs sensitivity analysis for the ADE and "both" returns sensitivity analysis for the ACME and ADE.

The tabular output from the summary function displays the values of ρ at which the confidence intervals contain zero for the ACME. For both the control and treatment conditions, the confidence intervals for the ACME contain zero when ρ equals 0.3 and 0.4. An alternative

but mathematically equivalent way to conduct sensitivity is in terms of the product of R2 (or

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−0.5 0.0 0.5 −0.2 −0.1 0.0 0.1 0.2 Anxiety Sensitivity Parameter: ρ A v er

age Mediation Eff

ect 0 −0.5 0.0 0.5 −0.2 −0.1 0.0 0.1 0.2 Anxiety Sensitivity Parameter: ρ A v er

age Mediation Eff

ect

1

Figure 3: Graphical display of results from the medsens function. Results as a function of ρ.

in more detail elsewhere (Imai et al. 2010c,2011,2010a), the first row captures the point at

which the ACME is 0 as a function of the proportions of residual variance in the mediator and outcome explained by the hypothesized unobserved confounder. The second line uses the

total variance instead of residual variance. We use R∗2 for residual variance and ˜R2 for total

variance. For example, when the product of the original variance explained by the omitted confounding is .049 the point estimate for ACME would be 0.

A graphical display is often more intuitive and useful for the sensitivity analysis, especially

for the R2 _{interpretations. This can be done, as before, by passing the object from the}

medsens function to the plot function. The plot function allows the researcher to graphically summarize the results of sensitivity analysis either in terms of ρ (sens.par = "rho") or R2 statistics (sens.par = "R2").

R> plot(sens.out, sens.par = "rho", main = "Anxiety", ylim = c(-0.2, 0.2))

When using the R2 statistic version of sensitivity analysis the user must specify whether the

hypothesized confounder affects the mediator and outcome variables in the same direction or in different directions. This matters because the sensitivity analysis is in terms of the product

of R2 statistics. In the current example, we assume that the confounder influences both

variables in the same direction by setting sign.prod = "positive" (rather than sign.prod = "negative"). Here, we plot the total variance version of the sensitivity analysis. The bold

line represents the various combinations of the R2 statistics where the ACME would be 0

(in this case the product equals .049). The graphical display also presents the corresponding

contour plots for other products of the R2 statistics.

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ACME0(R ~ M 2 ,R~_Y2), sgn(λ2λ3)=1 −0.1 −0.05 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 R ~ M 2 R ~ Y 2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ACME1(R ~ M 2 ,R~_Y2), sgn(λ2λ3)=1 −0.1 −0.05 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 R ~ M 2 R ~ Y 2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Figure 4: Graphical display of results from the medsens function as a function of ˜R2.

4. Causal mediation analysis of multilevel data

As of version 4.2, the mediation package supports causal mediation analysis of multilevel data

via the lmer and glmer functions in the lme4 package (Bates, Maechler, Bolker, and Walker

2014). Researchers are often interested in analyzing data where individual observations such

as students, patients, and employees are clustered within groups such as schools, hospitals, and companies. Data on individuals may be correlated within groups, but also different groups may have different data generating processes. Multilevel models take into account such heterogeneity within and between groups simultaneously.

Mediation analysis of multilevel data can be categorized into various types depending on whether the treatment, mediator and outcome variables are each measured at the individual

or group level (seeKrull and MacKinnon 2001;Zhang, Zyphur, and Preacher 2009).

Regard-less of these types, researchers can use the mediate function to analyze multilevel data by choosing appropriate statistical models for the mediator and outcome variables. In this sec-tion, we illustrate the use of our package for multilevel data by focusing on two types of data structure: (1) the treatment is assigned at the group level whereas the mediator and outcome are measured at the individual level, and (2) both the treatment and mediator are group-level variables while the outcome is recorded at the individual level. Other combinations of data

levels can be handled via straightforward modifications to the syntax used in these examples.7

To illustrate the usage, we analyze data from the Education Longitudinal Study (2002)8_where

7

We note that as of the writing of this article the lme4 package is known to generate slightly different random number draws across computing platforms (Windows, Mac, etc.) for a given seed which given the simulation method used can generate small numerical differences in some cases.

8

To protect the anonymity of the individuals involved in the study, we generated hypothetical individual-level variables via multiple imputation. The results reported below do not take into account any statistical uncertainty due to the imputation procedure and should thus be regarded only as illustration. The original data can be obtained from Education Longitudinal Study (ELS), 2002: Base Year (ICPSR 4275) by the United States Department of Education, National Center of Education Statistics. http://www.icpsr.umich.edu/ icpsrweb/ICPSR/studies/4275.

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students are clustered within schools. The mediation package contains two related data sets. The student data set contains both student- and school-level variables organized at the student level. The school data set only contains school-level variables, such that the number of observations in this data set equals the number of unique levels of the school identifier variable (SCH_ID) in the student data set. As explained below in detail, the group-level data set (school) is required only when we analyze the data where both the treatment and the mediator are group-level variables.

4.1. Group-level treatment and individual-level mediator

First, consider the case where the treatment is a group-level variable but the mediator and outcome variables are measured at the individual level. In this case, we only need the student-level data set,

R> data("student", package = "mediation")

Here, we analyze as an example whether a school is Catholic or not (catholic) affects a student’s likelihood of fighting (fight) at the school, and hypothesize that a student’s emo-tional attachment to the school (attachment) functions as the causal mechanism. That is, we postulate that students in a Catholic school may have an increased sense of attachment to their school, which may in turn decrease their likelihood of getting involved in a fight. We model these causal processes using the following hierarchical logistic-normal regression model for the (binary) mediator,

P(Mij = 1) = logit−1 αj+ γ>Xij , αj = α + βTj + εj,

where i and j are student and school indicators, respectively, εj is a normally distributed

group-level stochastic error with mean zero, and Xij represents the vector of student-level

pre-treatment covariates (gender, income and pared). Likewise, we use the following model for the (binary) outcome,

P(Yij = 1) = logit−1 λj + φjMij + ζ>Xij , λj = λ + ψTj+ υj, φj = φ + θTj+ νj,

where υj and νj are group-level errors jointly bivariate normally distributed with mean zero.

If desired, more complex data generating processes can be assumed (with appropriate changes in the syntax for the models below), such as allowing for group-varying slopes on the treatment variable or incorporating group-level pre-treatment covariates.

Now, note that these two models can be equivalently written as follows,

P(Mij = 1) = logit−1 (α + εj) + βTj+ γ>Xij , and P(Yij = 1) = logit−1 (λ + υj) + ψTj+ (φ + νj)Mij+ θTjMij+ ζ>Xij , which can both be estimated via the glmer function,

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R> library("lme4")

R> med.fit <- glmer(attachment ~ catholic + gender + income + pared +

+ (1 | SCH_ID), family = binomial(link = "logit"), data = student)

R> out.fit <- glmer(fight ~ catholic * attachment + gender + income +

+ pared + (1 + attachment | SCH_ID), family = binomial(link = "logit"),

+ data = student)

These fitted objects can then be fed into the mediate function in the usual manner. R> med.out <- mediate(med.fit, out.fit, treat = "catholic",

+ mediator = "attachment", sims = 100)

R> summary(med.out)

Quasi-Bayesian Confidence Intervals Mediator Groups: SCH_ID

Outcome Groups: SCH_ID

Output Based on Overall Averages Across Groups

ACME (control) -0.00393 -0.00685 -0.00135 0

ACME (treated) -0.00392 -0.00777 -0.00142 0

ADE (control) -0.02564 -0.04068 -0.00605 0

ADE (treated) -0.02563 -0.03975 -0.00639 0

Total Effect -0.02956 -0.04432 -0.01182 0

Prop. Mediated (control) 0.12983 0.05464 0.31910 0

Prop. Mediated (treated) 0.12167 0.05060 0.36426 0

ACME (average) -0.00392 -0.00693 -0.00156 0

ADE (average) -0.02564 -0.04021 -0.00622 0

Prop. Mediated (average) 0.12575 0.05705 0.33895 0

The estimated mediation, direct, and total effects are all significantly different from zero. The results suggest that the school-level treatment (catholic) increases the value of the individual-level mediator (attachment), which in turn decreases the value of the outcome (fight), and also that the treatment decreases the value of the outcome directly or in different causal paths.

4.2. Group-level treatment and mediator

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the outcome is measured at the individual level. In this case, we need the second data set containing only the group-level variables,

R> data("school", package = "mediation")

Note that the group-level data set must also contain the group indicator used in the individual-level data set under the same variable name (SCH_ID in our running example). The current version of mediate also requires that the model frames of the mediator and outcome models contain the exact same set of groups, which becomes important when each model contains different covariates and some groups drop out of the model frames due to missingness. As an illustration, we investigate the effects of school-level economic condition (free; pro-portion of students who receive free lunch) on students’ tardiness (late; days per semester they are late for school). As a causal path, we postulate that school-level poverty negatively impacts school-level morale (smorale), which in turn increases tardiness among students. We use the following hierarchical regressions to model the hypothesized causal mechanism,

Mj = α + βTj + γ>Vj + εj, Yij = λj+ ζ>Xij+ υij,

λj = λ + θTj + φMj+ κ>Vj+ νj,

where Vj is the vector of school-level covariates (catholic and coed), Xij is the vector of

student-level covariates (gender, income and pared), and εj, υij and νj are each normally

distributed stochastic errors with mean zero. Again, more complex models can be used (e.g., adding a treatment-mediator interaction term to the outcome model) if desired.

In this case, the mediator model is solely composed of the school-level variables and fixed coefficients. Hence the mediator model can be fit via the lm function using the school-level data set,

R> med.fit <- lm(smorale ~ free + catholic + coed, data = school)

and the outcome model, which can be equivalently written as,

Yij = (λ + υj) + θTj+ φMj+ (γ>+ κ>)Vj+ ζ>Xij + υij, can be estimated with the lmer function and the student-level data set,

R> out.fit <- lmer(late ~ free + smorale + catholic + coed + gender +

+ income + pared + (1 | SCH_ID), data = student)

These fitted model objects can then be passed to the mediate function. Since the treatment variable is a continuous variable, we use the values of 3 and 4 as the control and treatment values, respectively, and estimate the quantities of interest in terms of these values.

R> med.out <- mediate(med.fit, out.fit, treat = "free", mediator = "smorale",

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Quasi-Bayesian Confidence Intervals Mediator Groups:

Outcome Groups: SCH_ID

Output Based on Overall Averages Across Groups

ACME 0.007094 0.002554 0.012211 0.00

ADE 0.020356 -0.000729 0.038802 0.06

Total Effect 0.027450 0.005651 0.047817 0.02

Prop. Mediated 0.260223 0.080815 0.851923 0.02

The estimated mediation effect is significantly different from zero, suggesting that the school-level treatment (free) decreases the value of the school-school-level mediator (smorale), which in turn increases the value of the outcome (late).

We conclude this section by providing more details about the current version of our package for multilevel mediation analysis. First, the summary function can produce estimates of group-specific effects by adding the output argument, which can be set to either "bygroup" or "byeffect". In the above example, summary(med.out, output = "bygroup") produces the output organized by schools, and summary(med.out, output = "byeffect") produces the output organized into quantities of interest. Group-specific effects can also be graphically displayed by plot(med.out, group.plots = TRUE). Second, the mediate function allows researchers to specify different groups in the mediator and outcome models (nested or non-nested). For example, it may be reasonable to assume that the mediator variable is correlated within schools but the outcome variable is clustered at the district level. In such a case, the group.out argument for the mediate function allows researchers to choose the group into which the estimated group-specific effects are aggregated.

The current version of the package also has some limitations for multilevel mediation analysis. First, it only allows for one group type for each model. For example, it is not possible to let coefficients of the mediator (or outcome) model vary not only for schools but also for districts. Second, the bootstrap-based uncertainty estimates for the mediation effects are not yet available. Third, the medsens function for sensitivity analysis cannot be applied to the mediate outputs based on multilevel regression models. Future updates may add these missing functionalities. Finally, it is important to reiterate that the validity of the estimates

crucially rests on Assumption 1, regardless of whether hierarchical models are fitted to the

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5. Design-based causal mediation analysis

An alternative approach to model-based inference is to use different research designs that are

specifically designed for identifying causal mechanisms. Imai et al. (2013) propose several

such designs and describe the assumptions required for the identification of causal mediation effects under each of the designs. In this section we briefly illustrate how to use our software to calculate the estimates of the quantities of interest under each design.

5.1. Single experiment design

The single experiment design randomizes the treatment variable and measures the mediating

and outcome variables. In Section3, we discussed estimation functions that can be used with

parametric and semi-parametric models. If the researchers wish to pursue a completely non-parametric approach the mediation package offers two options via the mediate.sed function. First, the researchers can continue to make the sequential ignorability assumption and non-parametrically estimate the ACME. This approach works only when the mediator variable is discrete. Second, the sharp bounds on the ACME can be computed under the assumption

that only the treatment is randomized. Imai et al.(2013) derive the bounds in the case with

all binary variables (treatment, mediator, and outcome) and show that, unfortunately, the bounds are never informative about the sign of the ACME (i.e., they always include 0). Most mediation analysis proceeds under the sequential ignorability assumption. Those

anal-yses also tend to be model-based, but they need not be. Imai et al.(2010c) outline a

design-based estimator for the ACME for when the mediator is discrete. This estimator for the ACME is fully nonparametric. One drawback to this estimator is that one can encounter mediator-treatment combinations for which there are no subjects because of data sparsity. Standard error calculation for this estimator is based on either the Delta method or the nonparametric bootstrap.

The mediate.sed function requires the names of the outcome, mediator, and treatment vari-ables, along with the name of the data frame that contains these variables. When SI = TRUE, the function will return the point estimates under the sequential ignorability assumption, and otherwise the results will be a set of sharp bounds for the ACME. The method for inference also differs slightly from the mediate function. When boot = TRUE the bootstrap is used, but when boot = FALSE, the Delta method is used to compute standard errors.

Below, we present an example using the framing data fromBrader et al.(2008). The treatment

variable is the same as before, i.e., treat, and the mediator is anx, which refers to a subject’s reported level of anxiety. This four level measure is one component of the emo variable that was previously used as the mediator and in the data all treatment-mediator combinations are present (a requirement for the estimator). The outcome variable in this example is english and measures on a four point scale how much someone supports English only laws, from strongly support to strongly oppose. Note that the mediate.sed function only takes numeric variables as arguments. Variables that are stored as factors must be converted to numeric variables as we show below.

R> framing$english <- as.numeric(framing$english) R> framing$anx <- as.numeric(framing$anx)

R> sed.est <- mediate.sed("english", "anx", "treat", data = framing,

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R> summary(sed.est)

Design-Based Causal Mediation Analysis

Single Experiment Design with Sequential Ignorability Confidence Intervals Based on Nonparametric Bootstrap

Estimate 95% CI Lower 95% CI Upper

ACME (control) 0.10212 -0.56766 1.011

ACME (treated) 0.07066 -0.21566 0.379

Sample Size Used: 265

The results from the summary function display the mediation effects along with the default

95% confidence intervals. In this example both ¯δ(0) and ¯δ(1) are not significantly different

from 0.

5.2. Parallel design

An alternative to the single experiment design is the “parallel design” proposed byImai et al.

(2013). In this design there are two separate experiments that are run in parallel with subjects

randomly assigned to one of the two experiments. The first experiment follows the single experiment design. In the second experiment, subjects are randomly assigned to treatment or control. Then, a randomly selected set of subjects in each condition is assigned a value of the mediating variable. A key assumption of this design is that the manipulation of the mediating variable is possible and has no direct effect on the outcome variable.

Under the parallel design, the ACME is not point identified without an additional assumption. The mediation package offers two options via the mediate.pd function. First, the researchers can assume no interaction between the treatment and mediating variables by setting NINT = TRUE. In this case, the mediate.pd function will calculate the ACME along with its boot-strap confidence intervals. Second, the assumption of no-interaction between treatment and mediator can be dropped via NINT = FALSE, and then the sharp bounds can be calculated for the ACME. These bounds may be informative about the sign (i.e., do not cover 0) and are always narrower compared to the bounds under the single experiment design where the only assumption is randomization of the treatment.

For illustration, we simulated data based on the media framing experiment by Brader et al.

(2008) by creating a population distribution of potential mediators and outcomes (see Imai

et al. (2013) for more details). We then sampled 1000 cases from this distribution. In this example, out represents the outcome variable (immigration attitudes), med represents the mediator (anxiety), and ttt represents the treatment. All variables are binary. The vari-able manip represents whether the subject had the mediator manipulated (−1 if mediator is manipulated down, 0 if no manipulation, and 1 if manipulated up). First, the no-interaction assumption is made and options for the number of bootstrap simulations and confidence inter-vals are specified. In this case, the mediation effect is estimated at −0.12 with 95% confidence intervals spanning [−0.21, −0.03]. In the second example, the no interaction assumption is

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dropped and the sharp bounds are calculated to span [−0.3, 0.3] for the control condition and [0.2, 0.77] for the treatment condition.

R> data("boundsdata", package = "mediation")

R> pd <- mediate.pd("out", "med", "ttt", "manip", boundsdata, NINT = TRUE,

+ sims = 100, conf.level = 0.95)

R> summary(pd)

Design-Based Causal Mediation Analysis

Parallel Design (with No Interaction Assumption) Estimate 95% CI Lower 95% CI Upper

ACME -0.1236 -0.2198 -0.035

R> pd1 <- mediate.pd("out", "med", "ttt", "manip", boundsdata,

+ NINT = FALSE)

R> summary(pd1)

Design-Based Causal Mediation Analysis Parallel Design (Interaction Allowed)

Lower Bound Upper Bound

ACME (control) -0.3207 0.330

ACME (treated) 0.2006 0.768

5.3. Parallel encouragement design

In many situations, perfect manipulation of the mediating variable may be difficult. In the parallel encouragement design, subjects are split into two separate experiments. The first experiment is based on the single experiment design. In the second experiment subjects are randomly assigned to the treatment and control conditions and then, within each condition, a subset of subjects are randomly encouraged to have a high or low value of the mediator. Both the mediator and outcome variable are then measured. The mediate.ped function reports the sharp bounds on two estimands. First is the ACME and second is the ACME for the “compliers” who respond to the encouragement. The calculation of these bounds is

accomplished via a standard linear programming approach as discussed inImai et al. (2013).

The parallel encouragement design requires the analyst to specifically design some form of encouragement. The functionality of the mediate.ped closely mirrors that of mediate.sed. The key difference is that the analyst must also include an indicator for encouragement. For

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We did this by creating a population distribution of potential mediators and outcomes, and compliance types. We then randomly draw the joint probabilities of the causal types and

assign an encouragement status for those in the encouragement condition (see Imai et al.

(2013) for more details). Based on the encouragement condition and encouragement status

(enc, −1 if mediator is encouraged down, 0 if no encouragement, and 1 if encouraged up), the observed binary values of the mediator (med.enc) and outcome (out.enc) are determined. Using this simulated data we can then pass it to the mediate.ped function for the parallel encouragement design.

R> data("boundsdata", package = "mediation")

R> ped <- mediate.ped("out.enc", "med.enc", "ttt", "enc", boundsdata) R> summary(ped)

Design-Based Causal Mediation Analysis Parallel Encouragement Design

Lower Bound Upper Bound

Population ACME (control) -0.43407 0.324

Complier ACME (control) -0.14649 0.208

Population ACME (treated) -0.02014 0.743

Complier ACME (treated) 0.01137 0.707

Here, the results from mediate.ped function are a set of sharp bounds. We see that for the compliers, the sharp bounds on ACME under the treatment condition are informative as they do not cross 0.

5.4. Crossover encouragement design

The fourth experimental design included in the mediation package is the crossover

encour-agement design. Under this design, subjects are exposed to two experiments, with each

subject participating in each experiment. In the first experiment, the treatment variable is randomized and the mediator and outcome variables observed. In the second experiment, the treatment condition is set to the opposite value from the first period, but an encouragement is given to a randomly selected set of subjects so that the mediator variable will take on the value observed in the first experiment. Under this design, the ACME is point identified for the set of subjects that are able to have their mediator value manipulated (known as “pliable units”). A crucial identification assumption is that the first experiment does not influence behavior in the second experiment. For this experimental design the mediate.ced function calculates point estimates and the bootstrap is used for estimates of uncertainty.

For illustration, we simulated data based on the identification assumptions necessary for this design. Y2 is the value of the outcome variable in the second experiment, M1 and M2 are the mediator values for the first and second experiment, T1 is the value of the treatment in the first experiment, and Z indicates whether the subject’s mediator value in the second experiment is encouraged to take on the value opposite to that observed in the first experiment. All variables are binary.

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R> data("CEDdata", package = "mediation")

R> ced <- mediate.ced("Y2", "M1", "M2", "T1", "Z", CEDdata, sims = 100) R> summary(ced)

Design-Based Causal Mediation Analysis Crossover Encouragement Design

Estimate 95% CI Lower 95% CI Upper

Pliable ACME (control) 0.09069 -0.11769 0.300

Pliable ACME (treated) 0.11935 -0.05790 0.313

The results from the mediate.ced function are point estimates and confidence intervals for the ACME under the treatment and control conditions. These estimates apply only to the pliable units. In this example, both values of the ACME are positive but the 95% confidence intervals overlap with zero.

6. Analysis of causally dependent multiple mechanisms

Our discussion so far has focused on a single mediator, M . Frequently, however, researchers take measurements for more than one mediating variable. Accounting for alternative mecha-nisms is indeed crucial for the identification of the mechanism of primary interest, especially when such mechanisms are causally not independent. This is because the alternative depen-dent mediators affect both the mediator of primary interest and the outcome variable, which,

by definition, violates the sequential ignorability assumption (Assumption1).

6.1. The methodology

Imai and Yamamoto(2013) develop methods for dealing with multiple mediators based on the current framework. We briefly review this framework. First, in the case of causally unrelated multiple mediators, it turns out that there is no need to fundamentally modify the current framework or estimation procedure. To see this, suppose that there are multiple causally unrelated mediators, and one is interested in estimating the causal mediation effects with respect to each of them. In this scenario, note that for each mediator, the other mediators are neither pre-treatment nor post-treatment confounders (since by construction they have no causal effect on the mediator of interest). Therefore, one can consistently estimate the desired effects by simply applying the mediate function successively for the mediators as explained

in Section 3, ignoring the existence of the other, causally unrelated, mediators each time.

Likewise, sensitivity analysis via the medsens function can be conducted for the mediators of interest in the usual fashion. The mediations function can be useful for such analysis. Second, when the multiple mediators are causally related (or equivalently, when one mediator acts as a post-treatment confounder for the other mediator on the outcome), we need to expand the notational framework, and the analysis requires new assumptions. Let Wi(t) denote the vector of the potential values of those alternative mediators given treatment status

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t. To allow the causal dependence of both the primary mediator and outcome on W , we

write the potential mediator and outcome as Mi(t, w) and Yi(t, m, w), respectively. The

observed values of these potential response variables can then be expressed as Wi = Wi(Ti),

Mi = Mi(Ti, Wi(Ti)), and Yi = Yi(Ti, Mi(Ti, Wi(Ti)), Wi(Ti)). The causal mediation effects can now be re-expressed using this notation as,

δi(t) = Yi(t, Mi(1, Wi(1)), Wi(t)) − Yi(t, Mi(0, Wi(0)), Wi(t)),

for t = 0, 1. Note that this quantity represents the treatment effects that are transmitted

through the mediator of primary interest Mi, irrespective of whether they also come through

the alternative mediators Wi or not. Therefore, the quantity of interest remains unchanged

from the previous sections, except that the existence of the other mediators are now explicitly taken into consideration.

The framework of Imai and Yamamoto (2013) is based on the following varying coefficient

linear structural equations model,

Mi(t, w) = α2+ β2it + ξ2i>w + µ>2itw + λ>2ix + ε2i, (8)

Yi(t, m, w) = α3+ β3it + γim + κitm + ξ>3iw + µ>3itw + λ>3ixi+ ε3i, (9) where E(ε2i) = E(ε3i) = 0 without loss of generality. Although these equations may resemble a traditional linear structural equations model at a first glance, they are considerably more flexible because the coefficients are all allowed to vary arbitrarily across individual units.

Imai and Yamamoto(2013) propose two strategies for the analysis of the average causal

medi-ation effects, ¯δ(t) ≡ E(δi(t)). First, it can be shown that the ACME is point identified under

the above model and sequential ignorability (a weaker version allowing for post-treatment

confounding; seeImai and Yamamoto) if the homogeneous interaction assumption is satisfied.

This additional assumption is formally written as,

Yi(1, m, Wi(1)) − Yi(0, m, Wi(0)) = Bi+ Cm

for any m. The assumption states that the degree of interaction between the treatment and the primary mediator is constant across individual units, which may or may not be plausible depending on the empirical context.

Second, when this assumption is violated, one can express the sharp bounds on the ACME as functions of a parameter representing the degree of the violation, and conduct a sensitivity analysis. The sensitivity parameter here is the standard deviation of the coefficient on the treatment-mediator interaction term, i.e.,

σ ≡ pVAR(κi),

and the expression for the sharp bounds are given in Imai and Yamamoto (2013, Footnote

6). Researchers can then analyze robustness to the potential violation of the homogeneous interaction assumption by examining how the location and width of the bounds vary as σ changes.

The sensitivity analysis can also be formulated in terms of two alternative sensitivity parame-ters, both based on coefficients of determination as in the single mediator case (see Section3.4). Specifically, we use the proportion of the residual or total variance of the outcome variable

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that would be explained by allowing the heterogeneity in the treatment-mediator interaction in the outcome model. These parameters are formally defined as

R2∗ = VAR(˜κiTiMi) VAR(η3i(Ti, Mi, Wi)) and Re2 = VAR(˜κiTiMi) VAR(Yi) , (10)

where ˜κi = κi− E(κi). Researchers may find these parameters to be easier to interpret in

substantive terms, as they represent how important it would be to incorporate the interaction

heterogeneity in order to explain the variation in the outcome model. Imai and Yamamoto

(2013) show that these parameters have a one-to-one relationship with σ, implying that the

ACME can also be written as a function of R2∗ or eR2.

6.2. Single experiment design

The above framework has been implemented in the mediation package as the multimed func-tion. The function takes a data frame containing the necessary variables (outcome, primary mediator, alternative mediator, treatment, and pre-treatment covariates if any) and outputs an object of class ‘multimed’, a list consisting of estimated bounds along with uncertainty es-timates. In the current version, only a single post-treatment confounder is allowed, although the theoretical framework accommodates more than one such confounder.

The functionality of multimed differs in important ways from mediate. First, there is not a separate function for sensitivity analysis. Instead, a sensitivity analysis is conducted within the function along with the estimates of the mediation effects. Second, the arguments for the multimed function are rather different. Here, the names of the outcome (outcome), first mediator (med.main), second mediator (med.alt) and treatment (treat) variables are passed to the function along with a vector of the names of the pre-treatment covariates to condition on (covariates). In the multimed function, inference can only be done with the nonparametric bootstrap.

To illustrate the use of the function we revisit the media framing example in Section 3.

Here, we use a different outcome variable immigr, which is a five category measure of whether immigration should be increased or decreased (treated as a continuous measure for the purpose of illustration). The main mediator is the same composite measure of anxiety, emo, and the treatment and pre-treatment covariates are defined as before. We now introduce an alternative mediator p_harm, which is an eight category measure of the perceived economic harm of immigrants. The reasoning behind the inclusion of this variable is that the media framing treatment may also affect participants’ opinion about immigrants by changing their factual belief about the economic impact of increased immigration, which may also affect the level of anxiety and therefore confound the mediator-outcome relationship.

R> Xnames <- c("age", "educ", "gender", "income")

R> m.med <- multimed(outcome = "immigr", med.main = "emo",

+ med.alt = "p_harm", treat = "treat", covariates = Xnames,

+ data = framing, sims = 100)

R> summary(m.med)

(28)

Estimates under the Homogeneous Interaction Assumption: Estimate 95% CI Lower 95% CI Upper

ACME (treated) 0.06447 -0.09734 0.23

ACME (control) 0.12397 0.01555 0.23

ACME (average) 0.10870 0.00618 0.21

Total Effect 0.41752 0.16818 0.62

Sensitivity Analysis:

Values of the sensitivity parameters at which ACME first crosses zero:

sigma(bounds) sigma(CI) R2s(bounds) R2s(CI) R2t(bounds) R2t(CI)

ACME (treated) 0.0299 0.0000 0.0300 0.0000 0.0178 0.00

ACME (control) 0.0489 0.0173 0.0800 0.0100 0.0474 0.01

ACME (average) 0.0423 0.0173 0.0600 0.0100 0.0356 0.01

The summary function produces two tables. The first table shows the estimated ACME and total treatment effect and their confidence intervals (default at 95%) under the homogeneous interaction assumption. Three variants of the ACME are shown: the ACME conditional on the treatment group, the control group, and the weighted average of the two with the weights being equal to the proportions of the treatment and control groups. The second table shows key summary results from the sensitivity analysis with respect to possible heterogeneity in treatment-mediator interactions. Specifically, the table presents the values of σ (column 1),

R2∗(column 3), and eR2(column 5) at which the estimated ACMEs equal zero. The remaining

columns (2, 4 and 6) show those values for the confidence bands of the three ACMEs. The results from the multimed function can also be analyzed graphically using the plot function. One can produce two types of plots, corresponding to the two tables in the summary output. First, one can plot the point estimates under the homogeneous interaction assumption by setting the type argument to "point", as shown below. The output is in Figure 5. R> plot(m.med, type = "point")

Second, the results from the sensitivity analysis with respect to σ, R2∗or eR2can be plotted. In this case, the type argument can be used to specify which parameter(s) the estimated ACME should be plotted against. The possible values are "sigma", "R2-residual", or "R2-total". One can also choose the types of the ACME from "treated", "control" and "average" via the tgroup argument. In the example below, we plot the estimated ACME for both treatment

and control conditions as a function of σ and eR2. The output is in Figure6.

R> plot(m.med, type = c("sigma", "R2-total"), + tgroup = c("treated", "control"))

6.3. Parallel design

Imai and Yamamoto (2013, Section 7) show that the above framework can also be applied

to the data collected under the parallel design. As discussed in Section 5.2, the parallel

design consists of two separate experiments to which subjects are randomly selected. In one experiment, only the treatment is randomized and the researcher observes the mediator and